Photo Doppler Problem

A friend and class-fellow of mine(B.B. Gupta) came up with the following interesting problem(during the starting days of our M.Sc.):-There's a photocell and a (monochromatic)source of light at a certain distance from the photocell.The source of light emits light of frequency lower than what is required for the photoelectric effect.So there is no photoelectric effect.Now the source of light starts moving towards the photocell with some constant velocity--the resultant doppler effect is sufficient to take the frequency beyond the threshold(in photocell's frame).The question is 'would the photoelectric effect now be observed?'If yes,where does the photon get the extra energy from?

This is the problem in its original form.Let us see what ideas are generated here.

A friend and class-fellow of mine(B.B. Gupta) came up with the following interesting problem(during the starting days of our M.Sc.):-There's a photocell and a (monochromatic)source of light at a certain distance from the photocell.The source of light emits light of frequency lower than what is required for the photoelectric effect.So there is no photoelectric effect.Now the source of light starts moving towards the photocell with some constant velocity--the resultant doppler effect is sufficient to take the frequency beyond the threshold(in photocell's frame).The question is 'would the photoelectric effect now be observed?'If yes,where does the photon get the extra energy from?

This is the problem in its original form.Let us see what ideas are generated here.

J.Singh

I'll take a guess at this. A light emitter will experience force in the direction opposite to its motion. This is the same as a rocket accelerating due to expelling fuel. The extra energy will come from the work done by the external agent which keeps the emitter moving at constant speed.

A friend and class-fellow of mine(B.B. Gupta) came up with the following interesting problem(during the starting days of our M.Sc.):-There's a photocell and a (monochromatic)source of light at a certain distance from the photocell.The source of light emits light of frequency lower than what is required for the photoelectric effect.So there is no photoelectric effect.Now the source of light starts moving towards the photocell with some constant velocity--the resultant doppler effect is sufficient to take the frequency beyond the threshold(in photocell's frame).The question is 'would the photoelectric effect now be observed?'If yes,where does the photon get the extra energy from?

This is the problem in its original form.Let us see what ideas are generated here.

J.Singh

Firstly, yes, the photoelectric effect would be observed.

The energy of a photon depends on the frame one is in. When one switches frames, there is no reason to believe that the energy of the photon should remain the same. Conservation of energy says that the energy of a system in any inertial frame is constant, but it does not say that the energy remains unchanged when one switches frames.

Consider a ball, for instance. In one frame it is moving slowly (or even stationary), in another frame it is moving rapidly. There is no reason to think that the ball's energy does not change when one changes frames -- and the same is true for photons. The only difference is that photons do have a constant velocity. However, while the velocity of a photon is always constant, its energy depends on the frame of reference.

I'll take a guess at this. A light emitter will experience force in the direction opposite to its motion. This is the same as a rocket accelerating due to expelling fuel. The extra energy will come from the work done by the external agent which keeps the emitter moving at constant speed.

Pete

It's during the period of acceleration(from rest to a constant velocity) that work is done on the photon(in the frame of the photocell)--since its speed is constant,the only thing that can increase is its frequency.In fact,one can derive the relativistic doppler formula from the work done--try doing this!

The energy of a photon depends on the frame one is in. When one switches frames, there is no reason to believe that the energy of the photon should remain the same. Conservation of energy says that the energy of a system in any inertial frame is constant, but it does not say that the energy remains unchanged when one switches frames.

Quite right.Just to add to what you've said--when one switches frames slowly(so that there is a period of acceleration),one can account for the energy change(as mentioned in the above post).What happens,however,if the source of light is switched on after it has attained a constant velocity?Then I can't use the energy change(due to work done during the period of acceleration) idea anymore.

We have to now understand what's happening at the atomic level?Clearly if there's an increase in frequency of the photon there must be a corresponding change in the energy levels of the emitter.Can anyone here show that such a change in energy levels does take place(independent of what source is used)?

Quite right.Just to add to what you've said--when one switches frames slowly(so that there is a period of acceleration),one can account for the energy change(as mentioned in the above post).What happens,however,if the source of light is switched on after it has attained a constant velocity?Then I can't use the energy change(due to work done during the period of acceleration) idea anymore.

We have to now understand what's happening at the atomic level?Clearly if there's an increase in frequency of the photon there must be a corresponding change in the energy levels of the emitter.Can anyone here show that such a change in energy levels does take place(independent of what source is used)?

I don't think it's quite as simple as a change in the energy level of the emitter. A formerly spherically symmetrical problem has been changed into a non-symmetrical problem. (I suppose you could think of the atoms being "squished", like Lorentz did). But clearly, radiation in one direction must be blueshifted, in another direction it will be redshifted - the frequency of radiation emitted will depend on the direction.

As far as the details go, I'm not aware of them. I do recall, however, that the standard formulation of Schrodinger's wave equation is not Lorentz invariant, so I wouldn't expect it to be valid for a relativistically moving atom (even though it works fine for a stationary atom).

The Klein-Gordon equation is Lorentz invariant, but it's spin-0. (If you're dealing with photons, you need spin-1). So it's close, but not-quite-right.

I don't think it's quite as simple as a change in the energy level of the emitter. A formerly spherically symmetrical problem has been changed into a non-symmetrical problem. (I suppose you could think of the atoms being "squished", like Lorentz did).

If not a change in energy levels,then what else?I agree it's difficult to work out the change in energy levels.

But clearly, radiation in one direction must be blueshifted, in another direction it will be redshifted - the frequency of radiation emitted will depend on the direction.

Directional dependence of doppler effect is what makes the problem even more difficult.

To make the matter worse,let me ask you what happens in the frame of the source--when it was at rest,no photoelectric effect took place.But when it starts moving towards the photocell,the photoelectric effect takes place--in the frame of the source there's no change in the frequency of the emitted photon.So clearly,the workfunction of the photocell changes in the source's frame(and this change also has directional dependence).

As far as the details go, I'm not aware of them. I do recall, however, that the standard formulation of Schrodinger's wave equation is not Lorentz invariant, so I wouldn't expect it to be valid for a relativistically moving atom (even though it works fine for a stationary atom).

Of course,it's not Lorentz invariant.

The Klein-Gordon equation is Lorentz invariant, but it's spin-0. (If you're dealing with photons, you need spin-1). So it's close, but not-quite-right.

So what if it's spin 0.You are not dealing with photons here--you are talking of energy levels of the atom.

So what if it's spin 0.You are not dealing with photons here--you are talking of energy levels of the atom.

I remember now that what we want for this problem is the Dirac equation. That's spin 1/2 (not spin 1), but we are interested in solving for the state of the electron.

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The main point I'm trying to make is this. Given that we have a Lorentz invariant theory, it is a consequence of the Lorentz invariance of the theory that photons that are emitted isotropically in the rest frame of an atom must appear to be doppler shifted in a moving frame of reference according to the appropriate relativistic doppler shift laws.

Energy conservation, to my way of thinking, doesn't have anything directly to do with the issue. If you pick any inertial frame, energy will be conserved. Whatever energy the photon has ultimately has some source in the emitting mechanism.

So we can point to the Lorentz invariance of the theory as a justification for how we know it must turn out. Actually working out the details is not something I've done, though, in fact it took me a while even to name the right equation :-(. If you are really interested in the details, you might try the QM forum, though I think that you might well get a similar answer to what I wrote - that we know the soution must be Lorrentz invariant, and it's a lot easier to do the math in the rest frame of the atom.

So we can point to the Lorentz invariance of the theory as a justification for how we know it must turn out. Actually working out the details is not something I've done, though, in fact it took me a while even to name the right equation :-(. If you are really interested in the details, you might try the QM forum, though I think that you might well get a similar answer to what I wrote - that we know the soution must be Lorrentz invariant, and it's a lot easier to do the math in the rest frame of the atom.

It would be nice if you could make a copy of this thread in the QM forum so that we could have inputs from that forum too.

A moderator could move the thread, but I don't think it's possible to have the thread in more than one forum. Last I checked Doc Al and ZapperZ were our moderators, they would know more.

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Yes, Doc Al and ZapperZ are still our moderators. If they don't notice and reply to this thread, you might drop either or both of them an E-mail, asking them what is possible. I'm sure that they'll be willing to move the thread if that's what you want. I suspect that cross-posting is probably not allowed, but I don't know that for a fact.